The arithmetic sequence $a_i$ is defined by the formula: $a_1 = 4$ $a_i = a_{i - 1} + 11$ Find the sum of the first $650$ terms in the sequence.
Solution: Getting started Let's write out the first few terms of the series: $4 + 15 + 26 + 37...$ We're dealing with an arithmetic series because the difference between terms is constant. That is, each term is $11$ greater than the one before it. We need a formula to compute the sum of the terms. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = 4)$ and the number of terms $(n = {650})$ are given in the question. We need to find the last term $(a_n)$. Step 1: Find $a_n$ (the last term) There are $650 -1= 649$ terms after the first term. The sequence increases by $11$ for each new term. So, the sequence increases by a total of $649 \cdot 11 = 7139$ from where it starts at $4$. That means the last term must be $4+7139 = {7143}$. In other words, $a_n = {7143}$. Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{650}}&= \dfrac {\left({4} + {7143} \right)}{2} \cdot {650} \\\\ S_{{650}} &= 3573.5 \left(650\right) \\\\ S_{{650}} &= 2{,}322{,}775\end{aligned}$ The answer $ 2{,}322{,}775 $